## Some calculations of sigma_2

On four-manifold , we define Shouten tensor

and Einstein tensor and gravitational tensor

Suppose is the elemantary symmetric function

Thinking of as a tensor of type . is defined as applied to eigenvalues of . Then

Notice . Easy calculation reveals that

Under conformal change of metric , we have

We want to solve the equation , which is equivalent to solve

This is an fully nonlinear equation of Monge-Ampere type. Under local coordinates, the above equation can be treated as

where . This equation is elliptic if the matrix is positive definite. In order to find that matrix, we need the linearized operator

Using the elementary identity

for any fixed matrix and . Now plug in is Schouten tensor and . One can calculate them as

Then we get

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